Some Functional Properties on Cartan–Hadamard Manifolds of Very Negative Curvature

In this paper, we consider Cartan–Hadamard manifolds (i.e., simply connected, complete, of non-positive sectional curvature) whose negative Ricci curvature grows polynomially at infinity. We show that a number of functional properties, which typically hold on manifolds of bounded curvature, remain true in this setting. These include the characterization of Sobolev spaces on manifolds, the so-called Calderón–Zygmund inequalities and the Lp-positivity preserving property, i.e., u∈Lp&(-Δ+1)u≥0⇒u≥0. The main tool is a new class of first- and second-order Hardy-type inequalities on Cartan–Hadamard manifolds with a polynomial upper bound on the curvature. In the last part of the manuscript we prove the Lp-positivity preserving property, p∈[1,+∞], on manifolds with subquadratic negative part of the Ricci curvature. This generalizes an idea of B. Güneysu and gives a new proof of a well-known condition for the stochastic completeness due to P. Hsu.